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On quadrature rules, inequalities and error bounds. (English) Zbl 1179.41031

The author considers six operators approximating the integral mean value \(\frac{1}{2}\int_{-1}^{1}f(x)dx.\) All of them are connected with the very well known rules of approximate integration: Chebyshev quadrature, Gauss-Legendre quadrature with two and three knots, Lobatto quadrature with four and five knots and Simpson’s rule.
In this paper the order structure of the set of these six operators is established in the class of 5-convex functions. An error bound of the Lobatto quadrature rule with five knots is given for less regular functions as in the classical result.

MSC:

41A55 Approximate quadratures
26A51 Convexity of real functions in one variable, generalizations
26D15 Inequalities for sums, series and integrals
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